This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.

翻译：本文研究在比特最大后验（bit-MAP）译码下，Reed-Muller（RM）码经二进制无记忆对称（BMS）信道传输的性能。主要结论是：对于固定BMS信道，二进制RM码族能够在趋近信道容量的速率下实现比特差错概率趋于零。这一结果部分解决了连接信息论与纠错码领域的长期开放性难题。与早期针对二进制擦除信道的研究结果不同，新证明不依赖超收缩性，而是结合了RM码的嵌套性质与新的信息不等式，该不等式关联了广义外信息转移函数与外部最小均方误差。